I am looking for the relations and analogies between the Perelman's entropy functional,$\mathcal{W}(g,f,\tau)=\int_M [\tau(|\nabla f|^2+R)+f-n] (4\pi\tau)^{-\frac{n}{2}}e^{-f}dV$, and notions of entropy from statistical mechanics. Would you please explain it in details?

The more standard notions of entropy, notably Boltzmann and Shannon are roughly of the form $\int u\log u\,d\mu$ and if in Perelman's definiton you set $u = e^{-f}$ you get one term like this. The gradient term looks to me more like Fisher information, which can be viewed as the derivative of entropy with respect to time under Brownian motion. I suppose that the scalar curvature arises because everything is on a curved instead of flat space. The constant term arises from normalization. But this is all just my speculation.
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Deane YangMay 18 '13 at 14:43

Although it might seem like nothing more than a formal correspondence, the power of using entropy-type functionals for certain types of elliptic and parabolic PDE's indicates strongly to me that there is a deeper connection to the physical and information theoretic definitions of entropy than what we currently understand.
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Deane YangMay 18 '13 at 15:39

Perelman's entropy has the main term: $\int fe^{-f}d\mu,$ which is the
classical entropy with $u=e^{-f}$ as Deane Yang wrote. (Besides Section 5 of
Perelman, further discussion of entropy appeared later in some of Lei Ni's
papers as well as elsewhere.) Even though this term is lower order (in terms
of derivatives), geometrically it is the most significant as can be seen by
taking the test function to be the characteristic function of a ball
(multiplied by a constant for it to satisfy the constraint); technically, one
chooses a cutoff function. Thus Perelman proved finite time no local
collapsing below any given scale only assuming a local upper bound for $R,$
since the local lower Ricci curvature bound (control of volume growth is
needed to handle the cutoff function) can be removed by passing to the
appropriate smaller scale.

Heuristically (ignoring the cutoff issue), since the constraint is $\int(4\pi\tau)^{-n/2}e^{-f}d\mu=1,$ if
we take $\tau=r^{2}$ and $e^{-f}=c\chi_{B_{r}},$ then $c\approx\frac{r^{n}%
}{\operatorname{Vol}B_r}.$ So, if the time and scale are bounded from above, by Perelman's monotonicity, we have $$-C\leq\mathcal{W}(g,f,r^{2})\lessapprox r^{2}%
\max_{B_{r}}R+\ln\frac{\operatorname{Vol}B_r}{r^{n}},$$ yielding the volume ratio lower bound.

Perelman himself wrote about his entropy formula for the Ricci flow that "The interplay of statistical physics and (pseudo)-riemannian geometry occurs in the subject of Black Hole Thermodynamics, developed by Hawking et al. Unfortunately, this subject is beyond my understanding at the moment."

Perelman has given a gradient
formulation for the Ricci flow,
introducing an "entropy function"
which increases monotonically along
the flow. We pursue a thermodynamic
analogy and apply Ricci flow ideas to
general relativity. We investigate
whether Perelman's entropy is related
to (Bekenstein-Hawking) geometric
entropy as familiar from black hole
thermodynamics. From a study of the
fixed points of the flow we conclude
that Perelman entropy is not connected
to geometric entropy. However, we
notice that there is a very similar
flow which does appear to be connected
to geometric entropy. The new flow may
find applications in black hole
physics suggesting for instance, new
approaches to the Penrose inequality.

It seems helpful to look at the elliptic case first. As discovered by Lott-Villani and Sturm nonnegative Ricci curvature can be characterized by the property that the Boltzmann-entropy is convex along optimal transportation. This is very intuitive, imagine e.g. a pile of sand being transported from the south to the north-pole on the sphere.

The idea for the Ricci flow is similar (being a (super)Ricci flow can be viewed as parabolic version of having nonnegative Ricci curvature), but the details are a bit more complicated. The $W$-functional can be written as derivative of a suitable Boltzmann-entropy (see Section 5 in Perelman's first paper) and the monotonicity of $W$ can be interpreted as convexity of this entropy, see the above lecture notes for details.